The inverse Kazhdan-Lusztig polynomial of a matroid

TL;DR

This paper introduces the inverse Kazhdan-Lusztig polynomials for matroids, computes them for specific classes, and proposes conjectures about their properties, providing new tools and insights into matroid invariants.

Contribution

It defines inverse Kazhdan-Lusztig polynomials for matroids, computes them for boolean and uniform matroids, and derives a new formula for classical Kazhdan-Lusztig polynomials of uniform matroids.

Findings

Computed inverse Kazhdan-Lusztig polynomials for boolean and uniform matroids

Derived a new formula for Kazhdan-Lusztig polynomials of uniform matroids

Conjectured nonnegativity and log-concavity of inverse polynomial coefficients

Abstract

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomials for boolean matroids and uniform matroids. As an unexpected application of the inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the Kazhdan-Lusztig polynomials for uniform matroids. Similar to the Kazhdan-Lusztig polynomial of a matroid, we conjecture that the coefficients of its inverse Kazhdan-Lusztig polynomial are nonnegative and log-concave.